Peel and halve the garlic cloves and peel ginger before or after cutting into thin slices. Cut the chicken into bite sized pieces, trimming off any excess fat etc as needed.
Warm the vegetable and sesame oil in a wok or medium-large skillet/frying pan over a medium-low heat. Add the garlic and ginger and cook for a couple minutes so that you can see the garlic become paler and soften, but take care they don't brown or burn.
Move the garlic and ginger to one side then add the chicken. Spread it out in a single layer and if needed, increase the heat a little to sear the outside of the chicken. Turn the pieces of chicken regularly to brown gently all over. You don't want them to get too brown, just gently colored, but cook so the chicken pieces are no longer pink.
Add the Shaoxing wine, light and dark soy sauces and sugar and stir so everything mixes and the chicken is coated. Bring to a simmer then reduce the heat so the mixture continues to simmer, but not too strongly.
Cook, stirring now and then, for around 15 minutes until the cooking liquid has reduced down to become relatively syrupy. As it's becoming thicker, taste the sauce to check the seasoning is to your taste - if needed, add a little more Shaoxing wine and/or sugar if it seems too salty. Don't cook so far that you have almost nothing left - it will continue to reduce and thicken over time and as it cools. Add the Thai basil leaves and stir through so they wilt into the sauce, then serve over rice.